Online Traveling Salesman Problems with Service Flexibility

The traveling salesman problem is a well-known combinatorial optimization problem. We are concerned here with online versions of this problem defined on metric spaces. One novel aspect in this article is the introduction of a sound theoretical model to incorporate "yes-no" decisions on which requests to serve, together with an online strategy to visit the accepted requests. To do so, we assume that there is a penalty for not serving a request. Requests for visit of points in the metric space are revealed over time to a server, initially at a given origin, who must decide in an online fashion which requests to serve to minimize the time to serve all accepted requests plus the sum of the penalties associated with the rejected requests. We first look at the special case of the non-negative real line. After providing a polynomial time algorithm for the offline version of the problem, we propose and prove the optimality of a 2-competitive polynomial time online algorithm based on reoptimization approaches. We also consider the impact of advanced information (lookahead) on this optimal competitive ratio. We then consider the generalizations of these results to the case of the real line. We show that the previous algorithm can be extended to an optimal 2-competitive online algorithm. Finally we consider the case of a general metric space and propose an original c-competitive online algorithm, where c = root(17+ 5)/4 approximate to 2.28. We also give a polynomial-time (1.5 rho + 1)-competitive online algorithm which uses a polynomial-time rho-approximation for the offline problem. (C) 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 58(2), 137-146 2011